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Improving the Clark-Suen bound on the domination number of the Cartesian product of graphs. (English) Zbl 1367.05155
Summary: A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of W. E. Clark and S. Suen [Electron. J. Comb. 7, No. 1, Notes N4, 3 p. (2000; Zbl 0947.05056)], $$\gamma(G \square H) \geq \gamma(G) \gamma(H)/2$$, where $$\gamma$$ stands for the domination number, and $$G \square H$$ is the Cartesian product of graphs $$G$$ and $$H$$. In this note, we improve this bound by employing the 2-packing number $$\rho(G)$$ of a graph $$G$$ into the formula, asserting that $$\gamma(G \square H) \geq(2 \gamma(G) - \rho(G)) \gamma(H)/3$$. The resulting bound is better than that of Clark and Suen whenever $$G$$ is a graph with $$\rho(G) < \gamma(G)/2$$, and in the case $$G$$ has diameter 2 reads as $$\gamma(G \square H) \geq(2 \gamma(G) - 1) \gamma(H)/3$$.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C76 Graph operations (line graphs, products, etc.)
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##### References:
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